Non-equilibrium models for scalar dissipation

In many reacting flows, the reactants are introduced into the reactor with an integral scale L that is significantly different from the turbulence integral scale Lu. For example, in a CSTR, Lu is determined primarily by the actions of the impeller. However, is fixed by the feed tube diameter and feed flow rate. Thus, near the feed point the scalar energy spectrum will not be in equilibrium with the velocity spectrum. A relaxation period of duration on the order of xu is required before equilibrium is attained. In a reacting flow, because the relaxation period is relatively long, most of the fast chemical reactions can occur before the equilibrium model, (4.93), is applicable. 

Despite the explicit dependence on Reynolds number, in its present form the model does not describe low-Reynolds-number effects on the steady-state mechanical-to-scalar time-scale ratio (R defined by (3.72), p. 76). In order to include such effects, they would need to be incorporated in the scalar spectral energy transfer rates. In the original model, the spectral energy transfer rates were chosen such that R(t) - Ro = 2 for Sc = 1 and V p = 0 in stationary turbulence. In the version outlined below, we allow Rr, to be a model parameter. DNS data for 90 R-,. suggest that R , is nearly constant. However, for lower 

Reynolds numbers, its value is significantly smaller than the high-Reynolds-number limit. Despite its inability to capture low-Reynolds-number effects on the steady-state scalar dissipation rate, the SR model does account for Reynolds-number and Schmidt-number effects on the dynamic behavior of R(t). 

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The material covered in the appendices is provided as a supplement for readers interested in more detail than could be provided in the main text. Appendix A discusses the derivation of the spectral relaxation (SR) model starting from the scalar spectral transport equation. The SR model is introduced in Chapter 4 as a non-equilibrium model for the scalar dissipation rate. The material in Appendix A is an attempt to connect the model to a more fundamental description based on two-point spectral transport. This connection can be exploited to extract model parameters from direct-numerical simulation data of homogeneous turbulent scalar mixing (Fox and Yeung 1999). 

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